Your trainer is probably one of the following two- very experienced and "can do everything at once" and because he is impatient he expects you to do it his way to hurry it all up; or fairly inexperienced as a trainer.
Either way, they don't realise effective ways of teaching people new to the role you are in now. From my own personal experience, I wasn't good at training when I first started it, the temptation is to try and tell them how/when to do things only because you personally would have done that task by then. Flying experience does not equal being a good trainer; it is a skill that must be learnt and refined.
Anyway, re HDG bug. Bug first then turn. The reasons for it have already been mentioned. I would suggest you continue to bug first. Note that there is nothing "wrong" with turning first then bugging, it's in most people's opinion not the best method.
When your trainer mentions it again, bring it up in the debrief. "I bug first to keep my level of situational awareness up. I haven't been taught your method before, what are the benefits? I'll give it a go and see if it suits me." Of course, we all know that you won't change, but you'll at least challenge their method in a good way and might give them an ego boost - it'll give them the chance to try and get their point across. Then they might accept if you don't change, or they'll shut up about it.
Re sector entry timing.
You are both correct, kind of. Outbound timing begins "over or abeam the fix, whichever is later." Your instructor may have forgotten this and just does it over the fix every time. For your sector 1/parallel, just hit the timer as you go overhead, bug, then turn. By the time you've done this, 5 or so seconds will have passed before you turn to parallel the inbound track which gives you more space when turning inbound. Then when the minute is up, bug, then turn inbound. The timing doesn't need to be millisecond accurate. I hate the saying but it will highlight my point "near enough is good enough".
Here's another point in relation to parallel joins that a fair few people forget. You don't have to intercept the inbound, you can turn in and just track direct to the aid. From the AIP "the aircraft is turned... to intercept the inbound track
or return to the fix" Put needle on the nose, keep it easy
